About

The determinant is a scalar value that describes the scaling factor of the linear transformation described by a matrix. It is essential in linear algebra for solving systems of linear equations (Cramer's Rule), finding inverses, and determining eigenvalues. A determinant of zero (det(A) = 0) indicates a singular matrix, meaning it is not invertible.

This tool supports square matrices from dimensions 2x2 up to 5x5. It employs Laplace Expansion (cofactor expansion) for calculation. While computationally heavier than Gaussian elimination, this method is superior for educational purposes as it breaks the problem down into smaller minor matrices, clearly showing the recursive structure of the calculation.

Formulas

For a square matrix A, the determinant is defined recursively. For a 2x2 matrix:

detabcd = ad − bc

For larger matrices (Laplace Expansion along the first row):

det(A) = n∑j=1 (-1)^1+j a_1j M_1j

Where M_1j is the determinant of the submatrix formed by removing row 1 and column j.

Reference Data

Dimension	Formula / Method	Complexity
2x2	ad − bc	O(1)
3x3	Sarrus' Rule or Expansion	O(n!)
NxN	Laplace Expansion (Recursive)	O(n!)
Properties	det(AB) = det(A)det(B)	Identity

Frequently Asked Questions

You can paste a comma-separated string of numbers. For a 3x3 matrix, provide 9 numbers separated by commas (e.g., '1,2,3,4,5,6,7,8,9'). The tool automatically fills the grid row by row.

Yes, decimal inputs are fully supported. The result will be calculated with standard JavaScript floating-point precision.

A determinant of zero means the matrix is "singular". Geometrically, this implies the transformation squashes the space into a lower dimension (e.g., a 3D volume becomes a flat 2D plane or line), and the matrix has no inverse.

The Laplace Expansion method used here has factorial complexity O(n!). Calculating a 5x5 matrix involves 5 separate 4x4 calculations, each involving 4 separate 3x3s, and so on. Going beyond 5x5 in a browser environment using this specific recursive algorithm can cause performance lags.